A385599 -id:A385599 - cp's OEIS frontend

A385182 Values of u in the quartets (1,u,v,w); i.e., values of u for solutions to (1+u) = v*(v+w), in positive integers, with v>1, sorted by nondecreasing values of u; see Comments.

5, 7, 9, 11, 11, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 23, 25, 26, 27, 27, 29, 29, 29, 31, 31, 32, 33, 34, 35, 35, 35, 37, 38, 39, 39, 39, 41, 41, 41, 43, 43, 44, 44, 45, 47, 47, 47, 47, 49, 49, 50, 51, 51, 53, 53, 53, 54, 55, 55, 55, 56, 57, 59, 59

Offset: 1

Clark Kimberling, Jun 23 2025

nonn

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m

If m is a prime, then (u,v,w) = (m+2,m+1,m-1) is the first solution (in the defined ordering of triples).

u >= 1 appears A056924(u+1)-1 times. - Pontus von Brömssen, Jul 06 2025

			First 30 quartets (1,u,v,w):
  m    u    v    w
  1    5    2    1
  1    7    2    2
  1    9    2    3
  1   11    2    4
  1   11    3    1
  1   13    2    5
  1   14    3    2
  1   15    2    6
  1   17    2    7
  1   17    3    3
  1   19    2    8
  1   19    4    1
  1   20    3    4
  1   21    2    9
  1   23    2   10
  1   23    3    5
  1   23    4    2
  1   25    2   11
  1   26    3    6
  1   27    2   12
  1   27    4    3
  1   29    2   13
  1   29    3    7
  1   29    5    1
  1   31    2   14
  1   31    4    4
  1   32    3    8
  1   33    2   15
  1   34    5    2
  1   35    2   16
1*(1+23) = 2*(2+10) = 3*(3+5) = 4*(4+2), so three of the rows are (1,23,2,10), (1,23,3,5), and (1,23,4,2).

Guide to related sequences:
  m |    u    |    v    |    w
  --+---------+---------+--------
  1 | A385182 | A385183 | A385184
  2 | A385592 | A385593 | A385594
  3 | A385595 | A385596 | A385597
  4 | A385598 | A385599 | A385600
  --+---------+---------+--------
Cf. A056924.

Clear[solnsM];
solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
rhs = GatherBy[rhs, First];
Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
If[Length[matching] > 0, Do[AppendTo[ans,
Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
Length[matching]}]], {u, max}];
ans = Flatten[ans, 1];
Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
Sort[Select[DeleteDuplicates[
ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[1, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385182 *)
Map[#[[3]] &, solns]    (* v, A385183 *)
Map[#[[4]] &, solns]    (* w, A385184 *)
(*Peter J.C.Moses, Jun 15 2025*)

A385598 The u sequence in quartets (4,u,v,w); i.e., values of u for solutions to 4(4+u) = v(v+w), in positive integers, v>m, sorted by nondecreasing values of u; see Comments.

Original entry on oeis.org

6, 8, 10, 11, 11, 14, 14, 16, 16, 17, 17, 18, 20, 20, 21, 22, 23, 23, 24, 24, 26, 26, 26, 26, 28, 29, 29, 30, 31, 31, 31, 32, 32, 32, 34, 35, 35, 36, 36, 36, 38, 38, 38, 38, 40, 40, 41, 41, 41, 41, 41, 42, 44, 44, 44, 45, 46, 46, 46, 47, 47, 48, 48, 50, 50

Offset: 1

Clark Kimberling, Jul 10 2025

nonn

A 4-tuple (m,u,v,w) is a quartet if m,u,v,w are positive integers such that m>v and and m*(m+u) = v*(v+w), with the values of u in nondecreasing order. When there is more than one solution for given m and u, the values of v are arranged in increasing order. Here, m=4; for m=1, see A385182.

			First 30 quartets (4,u,v,w):
   m    u    v    w
   4    6    5    3
   4    8    6    2
   4   10    7    1
   4   11    5    7
   4   11    6    4
   4   14    6    6
   4   14    8    1
   4   16    5   11
   4   16    8    2
   4   17    6    8
   4   17    7    5
   4   18    8    3
   4   20    6   10
   4   20    8    4
   4   21    5   15
   4   22    8    5
   4   23    6   12
   4   23    9    3
   4   24    7    9
   4   24    8    6
   4   26    5   19
   4   26    6   14
   4   26    8    7
   4   26   19    2
   4   28    8    8
   4   29    6   16
   4   29   11    1
   4   30    8    9
   4   31    5   23
   4   31    7   13
4(4+16) = 5(5+11) = 8(8+2), so (4,16,5,11) and (4,16,8,2) are rows.

Cf. A385182, A385592, A385599, A385600.

Clear[solnsM];
solnsM[m_, max_] := Module[{ans = {}, rhs = {}, u, v, w, lhs, matching},
Do[Do[AppendTo[rhs, {v*(v + w), v, w}], {w, max}], {v, m*(m + max)}];
rhs = GatherBy[rhs, First];
Do[lhs = m*(m + u); matching = Select[rhs, #[[1, 1]] == lhs &];
If[Length[matching] > 0, Do[AppendTo[ans,
Map[{m, u, #[[2]], #[[3]]} &, matching[[1]]]], {i,
Length[matching]}]], {u, max}];
ans = Flatten[ans, 1];
Select[Union[Map[Sort[{#, RotateLeft[#, 2]}][[1]] &,
Sort[Select[DeleteDuplicates[
ans], {#[[1]], #[[2]]} =!= {#[[3]], #[[4]]} &]]]], #[[1]] == m &]];
TableForm[solns = solnsM[4, 140], TableHeadings -> {None, {"m", "u", "v", "w"}}]
aa = Flatten[solns]
Map[#[[2]] &, solns]    (* u, A385598 *)
Map[#[[3]] &, solns]    (* v, A385599 *)
Map[#[[4]] &, solns]    (* w, A385600 *)
(*Peter J.C.Moses, Jun 15 2025*)

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cp's OEIS Frontend

A385182 Values of u in the quartets (1,u,v,w); i.e., values of u for solutions to (1+u) = v*(v+w), in positive integers, with v>1, sorted by nondecreasing values of u; see Comments.

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